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Wavelet and Fourier methods for solving the sideways heat equation. (English) Zbl 0959.65107

The authors consider the inverse heat conduction problem: determine the temperature \(u(x,t)\), \(0\leq x \leq 1\) from the noisy measurements \(g_m (t), \|g_m(\cdot)- g(\cdot)\|_{L^2} \leq \varepsilon\), when \(u(x,t)\) satisfies \(u_{xx}=u_t, x\geq 0, t\geq 0; u(x,0)=0, x \geq 0; u(1,t)=g(t), t \geq 0; u |_{x \to \infty}\) bounded. Assume that a constant \(M\) is given such that \(\|u(0, \cdot)\|_{L^2} \leq M\). The paper discusses discrete stabilization algorithms based on replacing the time derivative in the equation by wavelet–based approximation or a Fourier approximation. The results of numerical tests are presented.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
80A23 Inverse problems in thermodynamics and heat transfer
35R30 Inverse problems for PDEs
35K05 Heat equation
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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